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Generalized binomial -leap method for biochemical kinetics incorporating both delay and intrinsic noise
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10.1063/1.2919124
/content/aip/journal/jcp/128/20/10.1063/1.2919124
http://aip.metastore.ingenta.com/content/aip/journal/jcp/128/20/10.1063/1.2919124

Figures

Image of FIG. 1.
FIG. 1.

Schematic representation of the DSSA implementation (Ref. 23). Here, refers to the waiting time until the next reaction is scheduled and is the delay time of reaction . The dotted arrows point to the time line indicating when a reaction is updated. The reaction is specified below the arrow. denotes a nondelayed and a delayed reaction. Nondelayed reactions are updated when triggered. In case of a delayed consuming reaction, its reactants and products are separately updated. This is marked by and , respectively. If time steps and reactions are drawn but then ignored, they are crossed out. Those steps are marked as gray dashed lines and the steps replacing an ignored step are marked as gray solid lines.

Image of FIG. 2.
FIG. 2.

Illustration of hes1 autoinhibition.

Image of FIG. 3.
FIG. 3.

Delta-notch signaling in a one-dimensional multicellular setting.

Image of FIG. 4.
FIG. 4.

Mean trajectory of the decaying-dimerizing model obtained from SSA runs.

Image of FIG. 5.
FIG. 5.

Single run of method 1 using -selection formula (a) with .

Image of FIG. 6.
FIG. 6.

Single runs of methods 2 and 3 using -selection formulas (b) and (a) with , respectively. The plots show the differences in the resulting dynamics in the numbers of monomeric Hes1.

Image of FIG. 7.
FIG. 7.

Mean molecular number of monomeric Hes1 obtained from runs of method 2 using -selection formula (a) for different .

Image of FIG. 8.
FIG. 8.

Mean molecular number of monomeric Hes1 obtained from runs of method 2 using -selection formula (a) for different . Single DSSA runs were not allowed.

Image of FIG. 9.
FIG. 9.

Mean molecular number of monomeric Hes1 obtained from runs of method 2 using -selection formula (b) for different .

Image of FIG. 10.
FIG. 10.

Mean molecular number of monomeric Hes1 obtained from runs of method 2 using -selection formula (b) for different . Single DSSA runs were not allowed.

Image of FIG. 11.
FIG. 11.

Mean molecular number of monomeric Hes1 obtained from runs of method 3 using -selection formula (a) for different .

Image of FIG. 12.
FIG. 12.

Mean molecular number of monomeric Hes1 obtained from runs of method 3 using -selection formula (a) for different . Single DSSA runs were not allowed.

Image of FIG. 13.
FIG. 13.

Mean molecular number of monomeric Hes1 obtained from runs of method 3 using -selection formula (b) for different .

Image of FIG. 14.
FIG. 14.

Mean molecular number of monomeric Hes1 obtained from runs of method 3 using -selection formula (b) for different . Single DSSA runs were not allowed.

Image of FIG. 15.
FIG. 15.

DSSA and -DSSA (method 3) dynamics of Delta mRNA and Delta protein in cell 3. The numbers of proteins are scaled by 0.05.

Image of FIG. 16.
FIG. 16.

DSSA and -DSSA (method 3) dynamics of Her1 mRNA and Her1 protein in cell 3. The numbers of proteins are scaled by 0.05.

Image of FIG. 17.
FIG. 17.

Sizes of -steps over time for single -DSSA runs (method 3) for two different values of . The single DSSA step mechanism was disabled.

Image of FIG. 18.
FIG. 18.

Her1 protein dynamics in cell 3 obtained from single -DSSA runs (method 3) for two different values of . The single DSSA step mechanism was disabled.

Tables

Generic image for table
Table I.

Some simple reactions and their corresponding propensities , stoichiometric coefficients , and maximum number of potential reaction events .

Generic image for table
Table II.

The error statistics are based on simulations for the SSA and all three -leap algorithms using the -selection formula (a) with . The maximum absolute errors and the maximum relative errors (per species) are calculated from the mean trajectories with 300 equidistant time points in [0,30]. The maximum absolute errors occur at in the molecular numbers of species for method 1 and at in all other cases. The maximal relative errors (relative to the expected molecular numbers) occur at in the molecular numbers of species for all three methods and at otherwise. The table also presents the mean error (the deviation from the SSA mean trajectory) at time point . For the SSA, mean and standard deviations of the molecular numbers of at are (2441.86,12 101.07,31 329.83) and (49.69,95.84,102.74), respectively. The smallest errors are highlighted. The maximal relative error values are rounded to the fourth decimal place; all other values are rounded to the second decimal place.

Generic image for table
Table III.

The statistics are obtained as described in Table II, using the -selection formula (a) with .

Generic image for table
Table IV.

The statistics are obtained as described in the caption of Table II, using the -selection formula (b) with for all three -leap methods.

Generic image for table
Table V.

The statistics are obtained as described in the caption of Table II, using the -selection formula (b) with for all three -leap methods.

Generic image for table
Table VI.

Parameters used for the Hes1-dimer model.

Generic image for table
Table VII.

Statistics of method 2: mean values over 100 simulation runs.

Generic image for table
Table VIII.

Statistics of method 3: mean values over 100 simulation runs.

Generic image for table
Table IX.

Mean simulation runtimes (averaged over 100 simulation runs) of methods 2 and 3. A single DSSA simulation takes about .

Generic image for table
Table X.

Parameters for the five-cell Her1-Her7 model. Parameter values are taken from Horikawa et al. (Ref. 21).

Generic image for table
Table XI.

Effect of for all Hill-type reactions in the Her1/Her7 model on and the number of reactions (delayed/nondelayed) for method 2 with -selection formula (a). Single DSSA steps are not allowed. Data are based on a single simulation over . As a point of reference, from a single DSSA there are 693 411 nondelayed and 708 017 delayed reactions for the same simulation time.

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/content/aip/journal/jcp/128/20/10.1063/1.2919124
2008-05-30
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Generalized binomial τ-leap method for biochemical kinetics incorporating both delay and intrinsic noise
http://aip.metastore.ingenta.com/content/aip/journal/jcp/128/20/10.1063/1.2919124
10.1063/1.2919124
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